In Problem (REF ), we parametrize the low-rank density matrix \(\rho \)  by its factors \(U \in \mathbb {C}^{d \times r}\)  instead of a \({d \times d}\)  Hermitian matrix and then imposing the low-rank constraint as in (REF ).
By rewriting \(\rho = UU^\dagger \) ,
both the PSD constraint
and the low-rank constraint
are automatically satisfied, leading to the unconstrained non-convex formulation in (REF ).
Moreover, working in the factored space improves time and space complexities [1]}, [2]}, [3]}, [4]}.
